When axion-like dark matter field interacts with nuclear spins, the axion-like dark matter can be considered as a time-oscillating pseudo-magnetic field directed along the \(\mathbf {v}\) . Thus, the dark-matter couplings to nuclear spins can be seen as a Zeeman interaction between nuclear spins and this pseudo-magnetic field generated by axion-like particles. The resultant Hamiltonian is [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}
\(\begin{aligned}\mathcal {H}_{\rm {spin}}\approx \textrm {g}_{\rm {aNN}} \sqrt{2 \hbar ^3 c \rho _{\rm {DM}}} \sin (2\pi \nu _{a}t) \mathbf {v} \cdot \textbf {I}_{\rm {N}},\end{aligned}\) 
